10 Sparse PK Sampling

10.1 What is sparse PK?

In dense PK, each subject contributes a full concentration-time profile (multiple samples per subject). In sparse PK, each subject contributes only one or a few samples, and the population-level profile is reconstructed by pooling across subjects.

Sparse sampling is common in:

Preclinical toxicokinetic studies — serial sacrifice designs where each animal is sampled at only one timepoint
Paediatric or special population studies — where blood volume limits the number of draws per patient
Large observational studies — with opportunistic sampling

PKNCA handles sparse PK by computing a mean concentration-time profile across all subjects at each timepoint, then running NCA on that mean profile, along with a standard error estimate from the Nedelman & Jia (1998) method.

10.2 Dataset: serial sacrifice design

We construct a synthetic dataset following the Holder et al. (1999) design: 6 nominal timepoints, 3 subjects sampled at each timepoint (18 subjects total, 1 sample each).

set.seed(42)

# True PK parameters (oral, 1-compartment)
ka  <- 1.5   # h⁻¹ absorption rate
kel <- 0.25  # h⁻¹ elimination rate
F   <- 1.0
dose <- 100  # mg
V   <- 50    # L

pk_conc <- function(t, ka, kel, dose, V) {
  (dose / V) * (ka / (ka - kel)) * (exp(-kel * t) - exp(-ka * t))
}

timepoints <- c(0, 0.5, 1, 2, 4, 8, 24)  # include t=0 (pre-dose, conc=0)
n_per_time <- 3

d_sparse <- lapply(timepoints, function(t) {
  true_c <- pk_conc(t, ka, kel, dose, V)
  data.frame(
    time    = t,
    conc    = if (t == 0) rep(0, n_per_time)
              else pmax(0, rnorm(n_per_time, mean = true_c, sd = true_c * 0.25)),
    Subject = paste0("S", sprintf("%02d", which(timepoints == t) * 10 + seq_len(n_per_time)))
  )
}) |> bind_rows()

head(d_sparse, 12)

   time      conc Subject
1   0.0 0.0000000     S11
2   0.0 0.0000000     S12
3   0.0 0.0000000     S13
4   0.5 1.3216758     S21
5   0.5 0.8453529     S22
6   0.5 1.0736708     S23
7   1.0 1.5446074     S31
8   1.0 1.4683935     S32
9   1.0 1.2982273     S33
10  2.0 1.8411027     S41
11  2.0 1.3045641     S42
12  2.0 2.0104313     S43

ggplot(d_sparse, aes(x = time, y = conc)) +
  geom_point(aes(colour = Subject), size = 3, show.legend = FALSE) +
  stat_summary(fun = mean, geom = "line", linewidth = 1, colour = "black") +
  stat_summary(fun = mean, geom = "point", size = 4, colour = "black", shape = 18) +
  labs(
    title = "Sparse PK — individual samples (coloured) and mean profile (black)",
    x = "Time (h)", y = "Concentration (mg/L)"
  ) +
  theme_minimal()

10.3 The sparse workflow

The only difference from standard NCA is sparse = TRUE in PKNCAconc(). Everything else is identical.

# Step 1: concentration object — flag as sparse
o_conc <- PKNCAconc(d_sparse, conc ~ time | Subject, sparse = TRUE)

# Step 2: dose object — one row per subject (all received the same dose at t=0)
d_dose <- data.frame(Subject = unique(d_sparse$Subject), dose = dose, time = 0)
o_dose <- PKNCAdose(d_dose, dose ~ time | Subject, route = "extravascular")

# Step 3: data object with sparse interval
o_data <- PKNCAdata(
  o_conc, o_dose,
  intervals = data.frame(
    start          = 0,
    end            = Inf,
    sparse_auclast = TRUE
  )
  # no imputation needed — t=0 pre-dose samples are in the dataset
)

# Step 4: run
o_nca <- pk.nca(o_data)
as.data.frame(o_nca)

# A tibble: 3 × 5
  start   end PPTESTCD       PPORRES exclude
  <dbl> <dbl> <chr>            <dbl> <chr>  
1     0   Inf sparse_auclast  10.4   <NA>   
2     0   Inf sparse_auc_se    0.644 <NA>   
3     0   Inf sparse_auc_df    5.58  <NA>

PKNCA returns three values for sparse AUC:

Parameter	Meaning
`sparse_auclast`	Mean AUC from 0 to last measurable time (Nedelman & Jia method)
`sparse_auc_se`	Standard error of the AUC estimate
`sparse_auc_df`	Degrees of freedom for confidence interval construction

10.4 How PKNCA computes sparse AUC

At each timepoint, compute the mean concentration across subjects (pooling all subjects sampled at that time)
Apply the lin up / log down trapezoidal rule to the mean profile to get AUClast
Estimate variance using the Nedelman & Jia (1998) formula, accounting for the covariance between AUC trapezoids via the Holder (2001) method
Return AUClast ± SE and degrees of freedom

# Inspect the mean profile PKNCA uses internally
mean_profile <- d_sparse |>
  group_by(time) |>
  summarise(mean_conc = mean(conc), n = n(), se = sd(conc) / sqrt(n()), .groups = "drop")

mean_profile

# A tibble: 7 × 4
   time mean_conc     n      se
  <dbl>     <dbl> <int>   <dbl>
1   0     0           3 0      
2   0.5   1.08        3 0.138  
3   1     1.44        3 0.0728 
4   2     1.72        3 0.213  
5   4     1.13        3 0.149  
6   8     0.276       3 0.0322 
7  24     0.00481     3 0.00146

10.5 BLQ handling in sparse PK

sparse = TRUE affects how BLQ values (coded as 0) are treated at each timepoint before averaging. The default BLQ treatment is controlled by PKNCA.options("conc.blq") and applies before the mean is computed.

Available mean methods (controlled internally):

"arithmetic mean" — simple mean of all values including BLQs as 0
"arithmetic mean, <=50% BLQ" — if strictly more than 50% of subjects at a timepoint are BLQ, the mean concentration at that timepoint is set to 0 (at exactly 50% BLQ the mean is not zeroed)

The second method is conservative: timepoints with majority-BLQ results don’t pull the mean above zero.

10.6 Serial sacrifice vs. repeated sampling

# Serial sacrifice: each subject contributes EXACTLY 1 sample
d_serial <- d_sparse  # our example above — each subject is at one timepoint only

# Mixed design: some subjects contribute multiple samples
# (PKNCA handles this too — subjects can appear at multiple times)
d_mixed <- bind_rows(
  d_sparse,
  data.frame(time = 4, conc = 0.8, Subject = "S11"),  # S11 sampled at 2 times
  data.frame(time = 8, conc = 0.3, Subject = "S11")
)

d_dose_mixed <- data.frame(Subject = unique(d_mixed$Subject), dose = dose, time = 0)
o_dose_mixed  <- PKNCAdose(d_dose_mixed, dose ~ time | Subject, route = "extravascular")
o_conc_mixed  <- PKNCAconc(d_mixed, conc ~ time | Subject, sparse = TRUE)
o_data_mixed  <- PKNCAdata(
  o_conc_mixed, o_dose_mixed,
  intervals = data.frame(start = 0, end = Inf, sparse_auclast = TRUE)
)
o_nca_mixed <- pk.nca(o_data_mixed)
# Note: degrees of freedom (sparse_auc_df) cannot be calculated
# when any subject contributes multiple samples — a current PKNCA limitation
as.data.frame(o_nca_mixed)

# A tibble: 3 × 5
  start   end PPTESTCD       PPORRES exclude
  <dbl> <dbl> <chr>            <dbl> <chr>  
1     0   Inf sparse_auclast  10.2   <NA>   
2     0   Inf sparse_auc_se    0.573 <NA>   
3     0   Inf sparse_auc_df    7.13  <NA>

Limitation: sparse_auc_df is NA when any subject contributes more than one sample per interval. This is a known current limitation of the Nedelman & Jia variance estimation in PKNCA.

10.7 Constructing a confidence interval

With AUC, SE, and df you can construct a t-based confidence interval:

res <- as.data.frame(o_nca) |>
  tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES)

auc <- res$sparse_auclast
se  <- res$sparse_auc_se
df  <- res$sparse_auc_df

if (!is.na(df) && !is.na(se)) {
  ci_lo <- auc - qt(0.975, df) * se
  ci_hi <- auc + qt(0.975, df) * se
  cat(sprintf("AUClast = %.2f  95%% CI: [%.2f, %.2f]\n", auc, ci_lo, ci_hi))
} else {
  cat(sprintf("AUClast = %.2f  SE = %.2f  (df unavailable for CI)\n", auc, se))
}

AUClast = 10.40  95% CI: [8.80, 12.00]

10.8 Key differences from dense NCA

Aspect	Dense PK	Sparse PK
`PKNCAconc` flag	`sparse = FALSE` (default)	`sparse = TRUE`
Profile per subject	Full individual profile	1–few samples only
NCA run on	Each subject’s own profile	Pooled mean profile
Result	One row per subject per parameter	One row per parameter (population level)
Half-life	✓ (per subject)	✗ (not available)
AUClast	✓	✓ (`sparse_auclast`)
SE of AUC	✗	✓ (`sparse_auc_se`)
Interval col name	`auclast`	`sparse_auclast`

10.9 Available sparse parameters

cols <- get.interval.cols()
sparse_params <- Filter(function(x) isTRUE(x$sparse), cols)
data.frame(
  parameter   = names(sparse_params),
  description = sapply(sparse_params, `[[`, "desc")
) |> knitr::kable()

	parameter	description
sparse_auclast	sparse_auclast	For sparse PK sampling, the area under the concentration time curve from the beginning of the interval to the last concentration above the limit of quantification
sparse_aumclast	sparse_aumclast	For sparse PK sampling, the area under the moment curve from the beginning of the interval to the last concentration above the limit of quantification
cl.sparse.last	cl.sparse.last	Clearance from sparse sampling calculated with population AUClast
mrt.sparse.last	mrt.sparse.last	Mean residence time from sparse sampling
vss.sparse.last	vss.sparse.last	Steady-state volume of distribution from sparse sampling
kel.sparse.last	kel.sparse.last	Elimination rate (as calculated from the MRTsparse.last)
vz.sparse.last	vz.sparse.last	Terminal volume of distribution from sparse sampling

10.10 Sparse AUMC and derived parameters (≥ 0.12.2)

PKNCA ≥ 0.12.2 adds sparse AUMC (area under the first-moment curve) and five new derived parameters, enabling MRT, clearance, and volume estimation from sparse data.

10.10.1 Sparse AUMC

Parameter	Description
`sparse_aumclast`	AUMC from the pooled mean sparse profile (0 → last measurable)
`sparse_aumc_se`	Standard error of the sparse AUMC estimate
`sparse_aumc_df`	Degrees of freedom for the AUMC variance estimate

Requesting sparse_aumclast = TRUE automatically computes sparse_auclast, sparse_auc_se, sparse_auc_df, sparse_aumc_se, and sparse_aumc_df as dependencies.

sparse_aumc_interval <- data.frame(
  start           = 0,
  end             = Inf,
  sparse_aumclast = TRUE   # AUC and SE are computed automatically as dependencies
)

o_nca_sparse_aumc <- pk.nca(PKNCAdata(o_conc, o_dose, intervals = sparse_aumc_interval))

as.data.frame(o_nca_sparse_aumc) |>
  select(PPTESTCD, PPORRES)

# A tibble: 6 × 2
  PPTESTCD        PPORRES
  <chr>             <dbl>
1 sparse_auclast   10.4  
2 sparse_auc_se     0.644
3 sparse_auc_df     5.58 
4 sparse_aumclast  43.1  
5 sparse_aumc_se    3.21 
6 sparse_aumc_df    3.92

10.10.2 Sparse-derived PK parameters

Once sparse AUC and AUMC are available, PKNCA can compute additional population-level parameters:

Parameter	Description
`mrt.sparse.last`	Mean residence time from sparse sampling
`cl.sparse.last`	Clearance: dose / sparse AUClast
`kel.sparse.last`	Elimination rate constant: 1 / MRT_sparse
`vss.sparse.last`	Steady-state volume of distribution from sparse sampling
`vz.sparse.last`	Terminal volume of distribution from sparse sampling

sparse_full_interval <- data.frame(
  start           = 0,
  end             = Inf,
  sparse_aumclast = TRUE,   # also pulls in sparse_auclast, SE, df
  mrt.sparse.last = TRUE,
  cl.sparse.last  = TRUE,
  kel.sparse.last = TRUE
)

o_nca_sparse_full <- pk.nca(PKNCAdata(o_conc, o_dose, intervals = sparse_full_interval))

as.data.frame(o_nca_sparse_full) |>
  filter(PPTESTCD %in% c("sparse_auclast", "sparse_aumclast",
                          "mrt.sparse.last", "cl.sparse.last", "kel.sparse.last")) |>
  select(PPTESTCD, PPORRES)

# A tibble: 5 × 2
  PPTESTCD        PPORRES
  <chr>             <dbl>
1 sparse_auclast   10.4  
2 sparse_aumclast  43.1  
3 cl.sparse.last    9.62 
4 mrt.sparse.last   4.15 
5 kel.sparse.last   0.241

vss.sparse.last and vz.sparse.last require a terminal elimination estimate (λz) which cannot be derived from the pooled mean sparse profile alone; they return NA unless additional terminal-phase information is available.

pkgdown reference: PKNCAconc() · PKNCAdose() · PKNCAdata() · pk.nca() · get.interval.cols()

--- title: "Sparse PK Sampling" --- ```{r setup, include=FALSE} library(PKNCA) library(dplyr) library(ggplot2) conflicted::conflicts_prefer(dplyr::filter, dplyr::select, .quiet = TRUE) ``` ## What is sparse PK? In **dense PK**, each subject contributes a full concentration-time profile (multiple samples per subject). In **sparse PK**, each subject contributes only one or a few samples, and the population-level profile is reconstructed by pooling across subjects. Sparse sampling is common in: - **Preclinical toxicokinetic studies** — serial sacrifice designs where each animal is sampled at only one timepoint - **Paediatric or special population studies** — where blood volume limits the number of draws per patient - **Large observational studies** — with opportunistic sampling PKNCA handles sparse PK by computing a **mean concentration-time profile** across all subjects at each timepoint, then running NCA on that mean profile, along with a standard error estimate from the Nedelman & Jia (1998) method. --- ## Dataset: serial sacrifice design We construct a synthetic dataset following the Holder et al. (1999) design: 6 nominal timepoints, 3 subjects sampled at each timepoint (18 subjects total, 1 sample each). ```{r} set.seed(42) # True PK parameters (oral, 1-compartment) ka <- 1.5 # h⁻¹ absorption rate kel <- 0.25 # h⁻¹ elimination rate F <- 1.0 dose <- 100 # mg V <- 50 # L pk_conc <- function(t, ka, kel, dose, V) { (dose / V) * (ka / (ka - kel)) * (exp(-kel * t) - exp(-ka * t)) } timepoints <- c(0, 0.5, 1, 2, 4, 8, 24) # include t=0 (pre-dose, conc=0) n_per_time <- 3 d_sparse <- lapply(timepoints, function(t) { true_c <- pk_conc(t, ka, kel, dose, V) data.frame( time = t, conc = if (t == 0) rep(0, n_per_time) else pmax(0, rnorm(n_per_time, mean = true_c, sd = true_c * 0.25)), Subject = paste0("S", sprintf("%02d", which(timepoints == t) * 10 + seq_len(n_per_time))) ) }) |> bind_rows() head(d_sparse, 12) ``` ```{r} ggplot(d_sparse, aes(x = time, y = conc)) + geom_point(aes(colour = Subject), size = 3, show.legend = FALSE) + stat_summary(fun = mean, geom = "line", linewidth = 1, colour = "black") + stat_summary(fun = mean, geom = "point", size = 4, colour = "black", shape = 18) + labs( title = "Sparse PK — individual samples (coloured) and mean profile (black)", x = "Time (h)", y = "Concentration (mg/L)" ) + theme_minimal() ``` --- ## The sparse workflow The only difference from standard NCA is `sparse = TRUE` in `PKNCAconc()`. Everything else is identical. ```{r} # Step 1: concentration object — flag as sparse o_conc <- PKNCAconc(d_sparse, conc ~ time | Subject, sparse = TRUE) # Step 2: dose object — one row per subject (all received the same dose at t=0) d_dose <- data.frame(Subject = unique(d_sparse$Subject), dose = dose, time = 0) o_dose <- PKNCAdose(d_dose, dose ~ time | Subject, route = "extravascular") # Step 3: data object with sparse interval o_data <- PKNCAdata( o_conc, o_dose, intervals = data.frame( start = 0, end = Inf, sparse_auclast = TRUE ) # no imputation needed — t=0 pre-dose samples are in the dataset ) # Step 4: run o_nca <- pk.nca(o_data) as.data.frame(o_nca) ``` PKNCA returns three values for sparse AUC: | Parameter | Meaning | |---|---| | `sparse_auclast` | Mean AUC from 0 to last measurable time (Nedelman & Jia method) | | `sparse_auc_se` | Standard error of the AUC estimate | | `sparse_auc_df` | Degrees of freedom for confidence interval construction | --- ## How PKNCA computes sparse AUC 1. At each timepoint, compute the **mean concentration** across subjects (pooling all subjects sampled at that time) 2. Apply the **lin up / log down** trapezoidal rule to the mean profile to get AUClast 3. Estimate **variance** using the Nedelman & Jia (1998) formula, accounting for the covariance between AUC trapezoids via the Holder (2001) method 4. Return AUClast ± SE and degrees of freedom ```{r} # Inspect the mean profile PKNCA uses internally mean_profile <- d_sparse |> group_by(time) |> summarise(mean_conc = mean(conc), n = n(), se = sd(conc) / sqrt(n()), .groups = "drop") mean_profile ``` --- ## BLQ handling in sparse PK `sparse = TRUE` affects how BLQ values (coded as 0) are treated at each timepoint before averaging. The default BLQ treatment is controlled by `PKNCA.options("conc.blq")` and applies **before** the mean is computed. Available mean methods (controlled internally): - **`"arithmetic mean"`** — simple mean of all values including BLQs as 0 - **`"arithmetic mean, <=50% BLQ"`** — if **strictly more than 50%** of subjects at a timepoint are BLQ, the mean concentration at that timepoint is set to 0 (at exactly 50% BLQ the mean is not zeroed) The second method is conservative: timepoints with majority-BLQ results don't pull the mean above zero. --- ## Serial sacrifice vs. repeated sampling ```{r} # Serial sacrifice: each subject contributes EXACTLY 1 sample d_serial <- d_sparse # our example above — each subject is at one timepoint only # Mixed design: some subjects contribute multiple samples # (PKNCA handles this too — subjects can appear at multiple times) d_mixed <- bind_rows( d_sparse, data.frame(time = 4, conc = 0.8, Subject = "S11"), # S11 sampled at 2 times data.frame(time = 8, conc = 0.3, Subject = "S11") ) d_dose_mixed <- data.frame(Subject = unique(d_mixed$Subject), dose = dose, time = 0) o_dose_mixed <- PKNCAdose(d_dose_mixed, dose ~ time | Subject, route = "extravascular") o_conc_mixed <- PKNCAconc(d_mixed, conc ~ time | Subject, sparse = TRUE) o_data_mixed <- PKNCAdata( o_conc_mixed, o_dose_mixed, intervals = data.frame(start = 0, end = Inf, sparse_auclast = TRUE) ) o_nca_mixed <- pk.nca(o_data_mixed) # Note: degrees of freedom (sparse_auc_df) cannot be calculated # when any subject contributes multiple samples — a current PKNCA limitation as.data.frame(o_nca_mixed) ``` > **Limitation:** `sparse_auc_df` is `NA` when any subject contributes more than one sample per interval. This is a known current limitation of the Nedelman & Jia variance estimation in PKNCA. --- ## Constructing a confidence interval With AUC, SE, and df you can construct a t-based confidence interval: ```{r} res <- as.data.frame(o_nca) |> tidyr::pivot_wider(names_from = PPTESTCD, values_from = PPORRES) auc <- res$sparse_auclast se <- res$sparse_auc_se df <- res$sparse_auc_df if (!is.na(df) && !is.na(se)) { ci_lo <- auc - qt(0.975, df) * se ci_hi <- auc + qt(0.975, df) * se cat(sprintf("AUClast = %.2f 95%% CI: [%.2f, %.2f]\n", auc, ci_lo, ci_hi)) } else { cat(sprintf("AUClast = %.2f SE = %.2f (df unavailable for CI)\n", auc, se)) } ``` --- ## Key differences from dense NCA | Aspect | Dense PK | Sparse PK | |---|---|---| | `PKNCAconc` flag | `sparse = FALSE` (default) | `sparse = TRUE` | | Profile per subject | Full individual profile | 1–few samples only | | NCA run on | Each subject's own profile | Pooled mean profile | | Result | One row per subject per parameter | One row per parameter (population level) | | Half-life | ✓ (per subject) | ✗ (not available) | | AUClast | ✓ | ✓ (`sparse_auclast`) | | SE of AUC | ✗ | ✓ (`sparse_auc_se`) | | Interval col name | `auclast` | `sparse_auclast` | --- ## Available sparse parameters ```{r} cols <- get.interval.cols() sparse_params <- Filter(function(x) isTRUE(x$sparse), cols) data.frame( parameter = names(sparse_params), description = sapply(sparse_params, `[[`, "desc") ) |> knitr::kable() ``` --- ## Sparse AUMC and derived parameters (≥ 0.12.2) PKNCA ≥ 0.12.2 adds sparse AUMC (area under the first-moment curve) and five new derived parameters, enabling MRT, clearance, and volume estimation from sparse data. ### Sparse AUMC | Parameter | Description | |---|---| | `sparse_aumclast` | AUMC from the pooled mean sparse profile (0 → last measurable) | | `sparse_aumc_se` | Standard error of the sparse AUMC estimate | | `sparse_aumc_df` | Degrees of freedom for the AUMC variance estimate | Requesting `sparse_aumclast = TRUE` automatically computes `sparse_auclast`, `sparse_auc_se`, `sparse_auc_df`, `sparse_aumc_se`, and `sparse_aumc_df` as dependencies. ```{r} sparse_aumc_interval <- data.frame( start = 0, end = Inf, sparse_aumclast = TRUE # AUC and SE are computed automatically as dependencies ) o_nca_sparse_aumc <- pk.nca(PKNCAdata(o_conc, o_dose, intervals = sparse_aumc_interval)) as.data.frame(o_nca_sparse_aumc) |> select(PPTESTCD, PPORRES) ``` ### Sparse-derived PK parameters Once sparse AUC and AUMC are available, PKNCA can compute additional population-level parameters: | Parameter | Description | |---|---| | `mrt.sparse.last` | Mean residence time from sparse sampling | | `cl.sparse.last` | Clearance: dose / sparse AUClast | | `kel.sparse.last` | Elimination rate constant: 1 / MRT_sparse | | `vss.sparse.last` | Steady-state volume of distribution from sparse sampling | | `vz.sparse.last` | Terminal volume of distribution from sparse sampling | ```{r} sparse_full_interval <- data.frame( start = 0, end = Inf, sparse_aumclast = TRUE, # also pulls in sparse_auclast, SE, df mrt.sparse.last = TRUE, cl.sparse.last = TRUE, kel.sparse.last = TRUE ) o_nca_sparse_full <- pk.nca(PKNCAdata(o_conc, o_dose, intervals = sparse_full_interval)) as.data.frame(o_nca_sparse_full) |> filter(PPTESTCD %in% c("sparse_auclast", "sparse_aumclast", "mrt.sparse.last", "cl.sparse.last", "kel.sparse.last")) |> select(PPTESTCD, PPORRES) ``` > `vss.sparse.last` and `vz.sparse.last` require a terminal elimination estimate (λz) which cannot be derived from the pooled mean sparse profile alone; they return `NA` unless additional terminal-phase information is available. --- ::: {.callout-note icon=false appearance="minimal"} **pkgdown reference:** [PKNCAconc()](https://humanpred.github.io/pknca/reference/PKNCAconc.html) · [PKNCAdose()](https://humanpred.github.io/pknca/reference/PKNCAdose.html) · [PKNCAdata()](https://humanpred.github.io/pknca/reference/PKNCAdata.html) · [pk.nca()](https://humanpred.github.io/pknca/reference/pk.nca.html) · [get.interval.cols()](https://humanpred.github.io/pknca/reference/get.interval.cols.html) :::